Question

Find the dot product f ·g on the interval[0, 1] for the functions f(x) = x, g(x) = x². A) 1 : 8

B) 1 : 6
C) 1 : 4
D) 1 : 3

Answer 1

The dot product of the functions f(x) = x and g(x) = x² on the interval [0, 1] is 1 : 4.

Step-by-step explanation:

To find the dot product of two functions f(x) = x and g(x) = x² on the interval [0, 1], we need to evaluate the integral of the product of the two functions over the given interval.

1. First, we multiply the two functions: f(x) * g(x) = x * x² = x³.
2. Next, we find the integral of the resulting function x³ over the interval [0, 1].
3. Using the power rule of integration, we integrate x³ as (1/4)x⁴.
4. Plugging in the upper and lower limits of integration, we evaluate the integral as ((1/4)(1)⁴) - ((1/4)(0)⁴) = 1/4 - 0 = 1/4.

Therefore, the dot product f · g on the interval [0, 1] is 1 : 4.